Properties of the symplectic structure of general relativity for spatially bounded space–time regions

Abstract

We continue a previous analysis of the covariant Hamiltonian symplectic structure of General Relativity for spatially bounded regions of spacetime.To allow for wide generality, the Hamiltonian is formulated using any fixed hypersurface, with a boundary given by a closed spacelike 2-surface. A main result is that we obtain Hamiltonians associated to Dirichlet and Neumann boundary conditions on the gravitational field coupled to matter sources, in particular a Klein-Gordon field, an electromagnetic field, and a set of Yang-Mills-Higgs fields. The Hamiltonians are given by a covariant form of the Arnowitt-Deser-Misner Hamiltonian modified by a surface integral term that depends on the particular boundary conditions. The general form of this surface integral involves an underlying "energy-momentum" vector in the spacetime tangent space at the spatial boundary 2-surface. We give examples of the * a = g ab t * b . Now, define a projection operator P B with respect to B by

Conserved quantities from the equations of motion: with applications to natural and gauge natural theories of gravitation

“…The (3+1) splitting of formula (45) can be found in [40] and for this reason we do not exhibit it here. In [4,9,19,31,40,50] the relations interplaying between the integrability conditions of (45) and boundary conditions on the metric and its derivatives are analysed in detail. We just point out that expression (45) with Dirichlet boundary conditions gives rise to the Brown-York quasilocal energy [31,40].…”

“…Let us consider a vertical vector field X on the configuration bundle, locally given as X = X i ∂ ∂y i = δy i ∂ ∂y i . By treating (8) as a new Lagrangian we can consider the variation δ X L ′ and, accordingly, we can make again use of the first variational formula (4). Notice that L ′ in general depends on the fields y i together with their derivatives up to some order h ≤ 2k, but it also depends on the variables ξ µ and ξ A together with their derivatives up to the orders r and s, respectively; see (9).…”

“…Just to mention a few of them we recall the Lagrangian method based on Noether's theorem [35,48,49,61,65], the Hamiltonian approach and the symplectic methods [4,6,11,14,17,19,44,53,63], the Hamilton-Jacobi-based techniques [10,13,15,16], and the formulations which are directly based on field equations [5,50,64]. These are just some of the many references which are most relevant for our purposes, whereby other important literature is quoted.…”

“…Roughly speaking the choice of such a direction corresponds to a control mode of the fields at the boundary which is based on physical grounds (e.g. micro-canonical or grandcanonical ensembles) and it is related to the asymptotic behaviour of specific solutions and to the specific physical observable one is aimed to determine (see, e.g., [4,9,16,19,40,50,53]). We then believe that, inside a formalism which we want to be as general as possible, both physical and mathematical reasons suggest to us that only variations δQ should be reasonably considered.…”

Hamiltonian boundary term and quasilocal energy flux

“…The associated energy flux expressions, calculated according to the respective prescriptions (30)(31)(32)(33), (presuming that £ Nθ = 0 = £ Nω , i.e. N is a Killing field of the reference geometry) take the form…”